On the index of a free-boundary minimal surface in Riemannian Schwarzschild-AdS

We consider the index of a certain non-compact free-boundary minimal surface with boundary on the rotationally symmetric minimal sphere in the Schwarzschild-AdS geometry with \(m>0\). As in the Schwarzschild case, we show that in dimensions \(n\ge 4\), the surface is stable, whereas in dimension three, the stability depends on the value of the mass \(m>0\) and the cosmological constant \(\Lambda <0\) via the parameter \(\mu :=m\sqrt{-\Lambda /3}\). We show that while for \(\mu \ge \tfrac{5}{27}\) the surface is stable, there exist positive numbers \(\mu _0\) and \(\mu _1\), with \(\mu _1<\tfrac{5}{27}\), such that for \(0<\mu <\mu _0\), the surface is unstable, while for all \(\mu \ge \mu _1\), the index is at most one.